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A330175
a(n) = n + floor(ns/r) + floor(nt/r), where r = sqrt(5) - 2, s = sqrt(5) - 1, t = sqrt(5).
2
15, 30, 46, 61, 78, 93, 109, 124, 141, 156, 172, 187, 204, 219, 235, 250, 267, 282, 297, 313, 328, 345, 360, 376, 391, 408, 423, 439, 454, 471, 486, 502, 517, 534, 549, 564, 580, 595, 612, 627, 643, 658, 675, 690, 706, 721, 738, 753, 769, 784, 801, 816, 832
OFFSET
1,1
COMMENTS
This is one of three sequences that partition the positive integers. In general, suppose that r, s, t are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are disjoint. Let a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked. Define b(n) and c(n) as the ranks of n/s and n/t. It is easy to prove that
a(n)=n+[ns/r]+[nt/r],
b(n)=n+[nr/s]+[nt/s],
c(n)=n+[nr/t]+[ns/t], where []=floor.
Taking r = sqrt(5) - 2, s = sqrt(5) - 1, t = sqrt(5) yields
FORMULA
a(n) = n + floor(ns/r) + floor(nt/r), where r = sqrt(5) - 2, s = sqrt(5) - 1, t = sqrt(5).
MATHEMATICA
r = Sqrt[5] - 2; s = Sqrt[5] - 1; t = Sqrt[5];
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t]
Table[a[n], {n, 1, 120}] (* A330175 *)
Table[b[n], {n, 1, 120}] (* A016789 *)
Table[c[n], {n, 1, 120}] (* A330176 *)
CROSSREFS
Sequence in context: A044840 A033012 A046046 * A072304 A188237 A190715
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jan 05 2020
STATUS
approved